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In recent years, considerable interest has been focused on the study of scattering from very rough surfaces. Very rough surfaces are defined as those with an rms height variation of the order of a wavelength and an rms slope of the order of unity. These surfaces occur in several practical problems including underwater acoustics, microwave scattering by terrain, ultrasound scattering by tissues, and optical scattering by rough metallic surfaces. In addition, recent experimental and numerical studies show backscattering enhancement phenomena from very rough surfaces.16 In spite of their theoretical interest and practical importance, theories of scattering by very rough surfaces are scarce7 and outside the range of validity of conventional theories such as the perturbation method and the Kirchhoff approximation.811 Recently, we proposed a theory based on the modified Kirchhoff approximation (KA) with angular and propagation shadowing for one—dimensional Dirichlet rough surfaces.12 This paper extends our previous theory to include scattering by very rough metallic and dielectric surfaces. The range of validity of the theory is examined by comparing it with the Monte Carlo simulation. Some material in this paper is also in our recent papers.2022 Numerical studies on very rough surfaces show that the second—order KA, when the surface integral is limited within the distance of double bounces without being intercepted by the surface, agrees well with the exact Monte Carlo simulation.13 This is consistent with the observation made by Liszka and McCoy that any signal which intersects the rough surface, one or more times, will be canceled by some higher iteration.14 This cancellation is accomplished by the shadowing function. This also indicates that the first— and second— order KA with proper shadowing gives a good approximate solution to very rough surface scattering. Conventional shadowing functions are used for first—order KA scattering. For second—order KA, in addition to the conventional shadowing, the shadowing representing the probability that the wave scattered from a point on the surface arrives at the other point on the surface without being intercepted by the surface is included. This shadowing is given by the angular and the propagation distance probabilities. These two shadowing functions, angular and propagation, modify the second—order KA and give the proper energy conservation and enhanced backscattering. The shadowing functions for the second—order KA used in this paper are similar to that used by Jin.15 The effect of shadowing for double scattering by random surfaces is also studied by Pavel'yev.16 Our analytical method employs the positive and negative traveling waves for the second— order KA and this results in a clear physical interpretation of the processes for the ladder and the cross or cyclic terms. The cyclic terms represent two waves propagating over the surface in opposite directions, giving rise to the backscattering enhancement. We also use Fourier transform in the vertical direction to facilitate computation of the second moments. Figure 1 shows the approximate range where our theory is applicable in terms of an rms height o and a correlation distance 1. The ranges where the Kirchhoff approximation, field perturbation, and phase perturbation methods are valid are noted by KA, FP and PP, respectively. The range where backscattering enhancement takes place is noted by E, and this is the range where none of the conventional techniques is applicable. Those points marked with fl are where our theory agrees well with exact numerical simulations, and the energy is conserved within 5 % error. Our theory covers most of the range E where the enhancement takes place, and the theory also reduces to conventional KA in the range where KA is applicable.
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